An air bubble of 1 cm radius is rising at a steady rate of `2.00ms^-1` through a liquid of density `1.5gcm^-3`. Neglect density of air. If `g=1000cms^-2`, then the coeffieciet of viscosity of the liquid is

Options

A. `0.166xx10^3` Poise
B. `1.66xx10^-3` poise
C. `166xx10^3` poise
D. `16.6xx10^3` poise

Explanation

To find the coefficient of viscosity of the liquid, we can follow these steps:

### Step 1: Identify the forces acting on the air bubble
The forces acting on the air bubble are:
- **Buoyant Force (Fb)**: This is the upward force exerted by the liquid on the bubble.
- **Viscous Force (Fv)**: This is the resistance force exerted by the liquid as the bubble moves upward.
- **Weight of the Bubble (W)**: Since we neglect the density of air, this force can be considered negligible.

### Step 2: Write the equation for buoyant force
The buoyant force can be calculated using the formula:
\[ F_b = \rho_l \cdot V \cdot g \]
where:
- \( \rho_l \) is the density of the liquid (1.5 g/cm³),
- \( V \) is the volume of the bubble,
- \( g \) is the acceleration due to gravity (1000 cm/s²).

The volume \( V \) of the bubble can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the bubble (1 cm).

### Step 3: Calculate the volume of the bubble
Substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (1 \text{ cm})^3 = \frac{4}{3} \pi \text{ cm}^3 \]

### Step 4: Substitute the volume into the buoyant force equation
Now substituting the volume into the buoyant force equation:
\[ F_b = \rho_l \cdot \left(\frac{4}{3} \pi r^3\right) \cdot g \]
Substituting \( \rho_l = 1.5 \text{ g/cm}^3 \), \( r = 1 \text{ cm} \), and \( g = 1000 \text{ cm/s}^2 \):
\[ F_b = 1.5 \cdot \left(\frac{4}{3} \pi (1)^3\right) \cdot 1000 \]

### Step 5: Calculate the viscous force
The viscous force can be expressed as:
\[ F_v = 6 \pi \eta r v \]
where:
- \( \eta \) is the coefficient of viscosity,
- \( r \) is the radius of the bubble,
- \( v \) is the velocity of the bubble (2 cm/s).

### Step 6: Set the buoyant force equal to the viscous force
At steady state, the buoyant force equals the viscous force:
\[ F_b = F_v \]
Thus,
\[ 1.5 \cdot \left(\frac{4}{3} \pi (1)^3\right) \cdot 1000 = 6 \pi \eta (1) (2) \]

### Step 7: Simplify and solve for \( \eta \)
Cancel \( \pi \) from both sides:
\[ 1.5 \cdot \frac{4}{3} \cdot 1000 = 12 \eta \]
Now, calculate the left side:
\[ 1.5 \cdot \frac{4}{3} = 2 \quad \text{and} \quad 2 \cdot 1000 = 2000 \]
So,
\[ 2000 = 12 \eta \]
Now, solving for \( \eta \):
\[ \eta = \frac{2000}{12} = \frac{500}{3} \approx 166.67 \text{ g/cm·s} \]

### Final Answer
The coefficient of viscosity of the liquid is approximately \( 166.67 \text{ g/cm·s} \).

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